How many squares are there on a chessboard? (In total - we're not asking you just to count the 64 single black and white squares!)
Once you think about this puzzle in the right way, it's not hard at all.
- There is one square that is the full size of the chessboard.
- There are 4 squares with sidelength one unit less than this (taking a single square of the chessboard as having one unit length).
- There are 9 squares with sidelength one unit less again, ie with sidelength 6 units.
- In general, there are n2 squares of sidelength [8-(n-1)].
Adding all these numbers, for n between 1 and 8, we see that there are
1 + 4 + 9 + 16 + 25 + ... + + 64 = 204
squares in total on the chessboard.
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